[Help] Is the number of entry evaluations recorded?

[chaozg]

Hi @maltezfaria,

I wanted to compare the cost of building a H-matrix vs using a simple low-rank approximation and was thinking of using the number of entry evaluations as an index. Do we keep a record of the number of entry evaluations in HMatrices.jl? Is there some other way to quantify the cost besides entry evaluations?

Regards,
Charlie

[maltezfaria]

This function computes something close to what you want I think:

function compression_ratio(H::HTypes)
    ns = Base.summarysize(H) # size in bytes
    nr = length(H) * sizeof(eltype(H))# represented entries
    return nr / ns
end

If you want the number of entries stored, you need to loop over all leaves, and add the number of entries in the dense blocks (m*n) and the number of entries in sparse blocks ((m+n)*r, where r is the rank). Currently, there is no function to do that.

[chaozg]

I see. Thanks for the hint.

[maltezfaria]

A small comment here: I don't really understand what you are trying to compare. If a matrix is low rank (or numerically low rank), I don't think you should use an HMatrix formalism. Hierarchical matrices are useful when you have full rank matrices containing large low-rank blocks right?

[chaozg]

Thank you for the follow-up. I actually do want to listen to your opinion!

• I completely agree with you if we know the matrix is low-rank a priori, there is no need to bother HMatrix.
• I'm thinking of a case that we don't know if the (high dimensional) matrix is low rank or not while we're interested in performing a matvec product with it. So my naive idea is to first pick low-rank approximation, e.g., partial pivoted Cholesky, and check the approximation accuracy in doing matvec with increasing ranks. Meanwhile, I can also use HMatrices and check HMatrice's performance in doing the same thing with decreasing atol in building the HMatrix. And this way we end up with a comparison of HMatrix and low-rank approximation on performing matvec.

Do you think this experiment makes any sense? Particularly, with HMatrices.jl, should I expect some convergence of the accuracy of 1) matrix approximation and 2) matvec product with decreasing atol?

[maltezfaria]

I think it will all depend on a priori knowledge of the matrix you are trying to approximate. In my domain (integral equations), much is know about the structure of the linear operator, such as the numerical low rank of blocks corresponding to "far" interactions. This is because the entries of the matrix are given by evaluating a kernel function that is "smooth" when the target and source points are sufficiently far apart. It is precisely this information that determines the admissibility condition for low-rank blocks. Without such knowledge, I am afraid you have to try and see if it works 😅. Perhaps we can discuss more details if you if you have a concrete problem or MWE.

[chaozg]

Thanks very much for the comment! I'll think about it and come back to you when I have a concrete/specific question